Decentralized Control of Discrete Event Systems with
Specializations to Local Control and Concurrent
Systems ∗
Shengbing Jiang and Ratnesh Kumar
Department of Electrical Engineering
University of Kentucky
Lexington, KY 40506-0046
{sjian0, kumar}@engr.uky.edu
Abstract
The decentralized supervisory control problem of discrete event systems under partial observation is studied in this paper. The main result of the paper is a necessary
and sufficient condition for the existence of decentralized supervisors for ensuring that
the controlled behavior of the system lies in a given range. The contribution of the
paper in relation to prior work is as follows: (i) Our setting of decentralized control
generalizes the prior ones; (ii) We present an alternative approach for solving the decentralized control problem, which leads to computational saving for concurrent systems
and certain other systems; (iii) Our generalized formulation and its solution lets us
extend several of the existing results reported in [1, 10, 6, 7, 11]. The results of our
paper are illustrated by an example of a simple manufacturing system.
Keywords: Discrete event system, supervisory control, controllability, observability,
decomposability, decentralized control.
1
Introduction
The supervisory control theory for discrete event systems was first proposed by Ramadge
and Wonham. In [8], they introduced the notion of controllability as a necessary and sufficient
condition for the existence of a supervisor that achieves a desired controlled behavior for a
given discrete event system under the complete observation of events. When the events
∗
This research was supported by the National Science Foundation under the Grant NSF-ECS-9709796.
are not completely observed by the supervisor but are filtered by an observation mask, an
additional condition of observability introduced by Lin and Wonham [5], and Cieslak et al.
[1] is needed for the existence of the supervisor. In the more general case of decentralized
control when there are more than one supervisors, the uncontrollable event set for the ith
supervisor is Σui , and its observation is filtered through the mask Mi , the condition of coobservability is required in place of observability, as shown Cieslak et al. [1] for the case
when the controlled behavior is given as a prefix-closed language, and later generalized to
the non-prefix-closed case by Rudie and Wonham [10].
In [6], Lin and Wonham considered a specialized version of the decentralized control,
where the mask Mi is a projection type mask, and the local controllable event set Σci is
a subset of local observable event set Σi := Mi (Σ). Under such assumptions, the authors
gave a sufficient condition in terms of the normality that guarantees that the decentralized
control can achieve the optimal behavior achievable by a centralized supervisor. Later in [7],
they extended their results to the systems with partial observation, where the local event
set Σi is the union of the local controllable event set and the local observable event set, i.e.,
Σi = Σci
S
Σoi , and the local supervisor Si is constructed by using only the local information
Ti (L(G)), where Ti is the projection mask from the global event set Σ to the local event set
Σi , and G is the state machine representing the discrete event system under control. They
gave a sufficient condition for the existence of decentralized supervisors that ensure that the
controlled behavior of the system lies in a given range expressed by local specifications.
In [11], Willner and Heymann also studied the problem of [6], except that the systems
they considered were concurrent systems. They introduced the notion of separability, and
under the assumption that Σui
T
Σj = ∅ for all i 6= j they proved that the separability is a
necessary and sufficient condition that guarantees that the decentralized control can achieve
the optimal behavior achievable by a centralized supervisor.
In this paper, we first examine the general decentralized control of concurrent systems
as studied in [7] (refer Figure 1), while relaxing the assumption that the local event set Σ i is
the union of the local controllable event set and the local observable event set. We obtain a
necessary and sufficient condition for the existence of decentralized supervisors that ensure
that the controlled behavior of the system lies in a given range, extending the result of [7]
where only a sufficient condition was given for a more constrained problem.
Next we consider the special case of decentralized local control, where the ith supervisor
is local when it observes events in its local set Σi and controls the local controllable events
Σci := Σc ∩ Σi . This is the special case of the decentralized control mentioned above, with
G
Tn
T1
set union
σ1
M (σ1)
1
Gn
.....
G1
σn
n
1
Σ disable Σ disable
S1
.....
M n(σ n)
Sn
Figure 1: Generalized decentralized control of concurrent systems
the masks Mi being the identity masks, and the controllable events satisfying the property
that Σci ∩ Σj ⊆ Σcj , i.e., an event controllable to any local supervisor is also controllable to
those supervisors which can observe it. It turns out that the condition of co-observability
can be replaced by the weaker condition of decomposability in this setting.
We also consider the decentralized control of concurrent systems with partial observations, which is a special case of the above problem, and satisfies L(G) = ∩i Ti−1 [L(Gi )],
where G =ki Gi . A necessary and sufficient condition is also derived for the existence of
decentralized supervisors.
Finally, an illustrative example based on a simple manufacturing system is provided.
The contribution of the paper is summarized as follows: (i) Our setting of decentralized
control generalizes the prior ones; (ii) We present an alternative approach for solving the
decentralized control problem, which leads to computational saving (this is made more precise
in Remark 2); (iii) Our generalized formulation and its solution lets us extend several of the
existing results as itemized below.
This paper extends the results of [1, 10, 6, 7, 11] on concurrent systems and partial
observations in the following ways:
1. The problems studied in [1, 10] are special cases of the problems considered in this
paper, and their results can be derived from our results as corollaries. For the concurrent systems, our results also result in computational savings in the synthesis of the
decentralized supervisors compared with the results of [1, 10].
2. In this paper, partial observation masks are introduced for local systems, which were
considered in [6, 11]. For the problem studied in [6, 11], we provide a necessary and
sufficient condition for the existence of decentralized supervisors, where [6] only gave a
sufficient condition, and [11] gave the same necessary and sufficient condition as ours
under a more restricted assumption.
3. For the problem studied in [7], we give a necessary and sufficient condition for the
existence of decentralized supervisors, where [7] only gave a sufficient condition in a
more restricted setting.
2
Background Review
This paper is set in the supervisory control framework for discrete-event systems devel-
oped by Ramadge and Wonham [8]. For the readers’ convenience, some background results
from the cited references are first provided in this section. For a detailed introduction of the
theory, readers may refer to [2].
An uncontrolled discrete event system is modeled as an automaton
G = (Q, Σ, δ, q0 , Qm ),
where Σ is a set of event labels, Q is a set of states, q0 ∈ Q is the initial state, Qm ⊆ Q is the
set of marked states, and δ : Σ × Q → Q, the transition function, is a partial function defined
at each state in Q for a subset of Σ. Let Σ∗ denote the set of all finite strings over Σ including
the null string . Then δ : Σ×Q → Q can be extended in the obvious way to δ : Σ∗ ×Q → Q.
The language generated by G is given by, L(G) := {s ∈ Σ∗ | δ(s, q0 ) is defined}, and the
language marked by G is given by, Lm (G) := {s ∈ Σ∗ | δ(s, q0 ) ∈ Qm }.
In general, a language over Σ is any subset of Σ∗ . The prefix-closure pr(K) of a language
K ⊆ Σ∗ is the set of all prefixes of strings in K. K is called prefix-closed if pr(K) = K.
To impose supervision on the system, the event set Σ is partitioned into two subsets Σ c
and Σu of controllable and uncontrollable events respectively. A supervisor S is a pair (R, ψ)
where R is an automaton which recognizes a language over the same event set as G, and ψ
is the feedback map from the event set and the states of R to the set {enable, disable}. If X
denotes the set of states of R, then ψ : Σ × X →{enable, disable} satisfies: ψ(σ, x) =enable
if σ ∈ Σu . R is considered to be driven by the strings generated by G, and in turn, at
each state of R, the control rule ψ(σ, x) dictates the occurrence of σ at the corresponding
state of G. The behavior of the supervised system is represented by an automaton S/G.
The language generated by the supervised system is denoted by L(S/G), and its marked
language is Lm (S/G) := L(S/G) ∩ Lm (G).
A supervisor S = ((X, Σ, ξ, x0 ), ψ) is complete for G if for every string s in Σ∗ and every
event σ in Σ, the conditions s ∈ L(S/G), sσ ∈ L(G), and ψ(σ, ξ(s, x0 )) =enable together
imply that sσ ∈ L(S/G). It will always be assumed that S is complete.
Given a nonempty prefix-closed sublanguage K of L(G), there exists a supervisor S such
that L(S/G) = K if and only if K is (L(G), Σu )-controllable, i.e., pr(K)Σu
T
L(G) ⊆ pr(K)
[8]. If K is not controllable, then a supervisor is synthesized for achieving the supremal
prefixed closed and controllable sublanguage of K, denoted by supP C(K).
When the supervisor can not observe all the events, the concept of observation mask is
S
introduced. An observation mask is a function M : Σ → ∆ {}, where 6∈ ∆, and ∆ is
called the set of observed events. The mask function can be extended to the set of strings
in a natural way. A supervisor S for G is said to be mask-compatible [2] if it observes only
M (L(G)). In this paper, it is assumed that all supervisors for partially observed systems
must be mask-compatible supervisors.
A prefix-closed sublanguage K of L(G) is said to be (L(G), M )-observable [5] if the
conditions s, s0 ∈ K, M (s) = M (s0 ), sσ ∈ K, and s0 σ ∈ L(G) together imply s0 σ ∈ K.
If K is not observable, then the infimal prefix closed and observable superlanguage of K
denoted inf P O(K) exists [5]. Given a nonempty prefix-closed sublanguage K of L(G) and
an observation mask M , there exists a supervisor S such that L(S/G) = K if and only if
K is (L(G), Σu )-controllable and (L(G), M )-observable [5]. Let A, E ⊆ Σ∗ be two languages
with A ⊆ E ⊆ L(G), M be an observation mask, then there exists a supervisor S such
that A ⊆ L(S/G) ⊆ E if and only if inf P CO(A) ⊆ E, where inf P CO(A) is the infimal
prefix-closed, controllable, and observable superlanguage of A [5].
3
Decentralized Control
In this section we study the general decentralized supervisory control problem of discrete
event systems with partial observations first introduced in [7]. We further generalize the
formulation of [7] by relaxing the assumption that the local event set is the union of the
local controllable event set and the local observable event set, and the observation mask is
the projection type. This assumption is obviously restrictive, as in a general setting there
may be local events that are neither locally controllable nor observable. Our formulation of
the problem is able to handle such cases.
Let G be a plant, Σ be the global event set, Σi ⊆ Σ, i ∈ I, be the local event sets. Let
Ti : Σ → Σi ∪ {} denote the natural projection from the global event set to the local event
set. We use Gi to denote the automaton with L(Gi ) = Ti [L(G)], i ∈ I, called the local
system; here L(Gi ) represents the ith local behavior of the system G. Let Σci ⊆ Σi be the
ith local controllable event set, then the global controllable event set is given by Σ c := ∪i Σci ,
the local uncontrollable event set is given by Σui := Σ − Σci , the global uncontrollable event
S
set is given by Σu := Σ − Σc = Σ − ∪i (Σ − Σui ) = ∩i Σui . Finally, let Mi : Σi → ∆i {}
be the local observation mask. If Si is a local supervisor for Gi , then we can extend Si to a
global supervisor S̃i for G, by letting S̃i not observe all events in Σ − Σi , and permanently
enable all events in Σ − Σci . Then it is easy to see that L(S̃i /G) = L(G)
T
Ti−1 [L(Si /Gi )],
i ∈ I.
Next we give the definition of decomposable introduced by Rudie-Wonham in [9, 10].
Decomposability is weaker than co-observability in general, and in the setting of decentralized
local control the two are equivalent [10].
Definition 1 For a language K ⊆ L(G), K is (G, {Ti , i ∈ I}) decomposable if K =
T
L(G) (∩i Ti−1 Ti (K)).
Consider the following “target specification” problem:
Given a nonempty prefix-closed sublanguage K of L(G), find local supervisors
{Si , i ∈ I} such that L((
V
i
T
S̃i )/G) = K, i.e., L(G) (∩i Ti−1 [L(Si /Gi )]) = K.
For the existence of the local supervisors, a necessary and sufficient condition is obtained in
the following theorem. Before giving the theorem, we first give a lemma which is needed in
the proof of the theorem. This lemma is similar to [11, Corollary 2.1] which applied to the
separability.
Lemma 1 Let G be a plant, Σ be the global event set, Σi ⊆ Σ, i ∈ I, be the local event
sets. Let Ti be the natural projection from Σ to Σi . A language K ⊆ L(G) is (G, {Ti , i ∈ I})
decomposable if and only if there exists a group of languages {Ki ⊆ Ti [L(G)], i ∈ I} such
T
that K = L(G) (∩i Ti−1 (Ki )).
Proof: The necessity is obvious. We only need to prove the sufficiency. Since K ⊆
L(G)
T
T
Ti−1 Ti (K), it suffices to show that K ⊇ L(G) (∩i Ti−1 [Ti (K)]). From hypothesis,
T
there exists a group of languages {Ki ⊆ Ti [L(G)], i ∈ I} such that K = L(G) (∩i Ti−1 (Ki )).
This implies
Ti (K) ⊆ Ti (L(G))
= Ti (L(G))
\ \
(
\
j
Ki
Ti Tj−1 (Kj ))
\ \
(
j6=i
⊆ Ki
Ti Tj−1 (Kj ))
Hence
L(G)
\
(∩i Ti−1 [Ti (K)]) ⊆ L(G)
\
(∩i Ti−1 (Ki )) = K.
This completes the proof.
Note when L(G) = Σ∗ , the condition of decomposability in view of Lemma 1 is equivalent
to the existence of {Ki ⊆ Ti (Σ∗ )} satisfying K = ∩i Ti−1 (Ki ). This last property was defined
as separability in [11].
Theorem 1 Let G be a plant, Σ be the global event set, Σi ⊆ Σ, i ∈ I, be the local event
sets. Let Ti be the natural projection from Σ to Σi , Σci ⊆ Σi be the local controllable event
set, Mi be the local observation mask. Given a nonempty prefix-closed sublanguage K of
L(G), there exists a group of local supervisors {Si , i ∈ I} such that L((
only if
K = L(G)
\
V
i
S̃i )/G) = K if and
(∩i Ti−1 [inf P CO i (Ti (K))]),
where S̃i is the extension of Si to the global system G, and inf P CO i (Ti (K)) is the infimal
prefix closed, (L(Gi ), Σui )-controllable and (L(Gi ), Mi )-observable superlanguage of Ti (K)
(Σui := Σ − Σci and L(Gi ) := Ti [L(G)] is the ith local behavior of the system).
Proof: The sufficiency is obvious, since we can choose the local supervisor Si such that
L(Si /Gi ) = inf P CO i (Ti (K)). Then the result follows from the definition of L((
V
i
S̃i )/G)
and the hypothesis.
To prove the necessity, we need Lemma 1. If the local supervisors {Si , i ∈ I} exist,
T
then K = L(G) (∩i Ti−1 [L(Si /Gi )]), which from Lemma 1 implies K is decomposable, i.e.,
T
K = L(G) (∩i Ti−1 [Ti (K)]). Since
Ti (K) = Ti [L(G)
\
(∩j Tj−1 [L(Sj /Gj )])]
⊆ Ti (L(G)) ∩ L(Si /Gi ) ∩j6=i Ti [Tj−1 (L(Sj /Gj ))]
⊆ L(Si /Gi ),
we have Ti (K) ⊆ inf P CO i (Ti (K)) ⊆ inf P CO i (L(Si /Gi )) = L(Si /Gi ), where the last
equality follows from the fact that L(Si /Gi ) is locally controllable and observable. Now that
K = L(G)
and
K = L(G)
\
\
(∩i Ti−1 [Ti (K)])
(∩i Ti−1 [L(Si /Gi )]),
we must have
K = L(G)
\
(∩i Ti−1 [inf P CO i (Ti (K))]).
This completes the proof.
Remark 1 If the projections {Ti , i ∈ I} in Theorem 1 are identity projections, i.e.,
∀i ∈ I, Σi = Σ, then the necessary and sufficient condition in Theorem 1 becomes K =
∩i [inf P CO i (K)]. From [3, Theorem 3], we know this is equivalent to K being (L(G), Σ u )controllable (Σu = ∩i Σui ), and (L(G), {Σui }, {Mi })-co-observable, which is the result obtained in [1, 10]. So Theorem 1 extends the result of [1, 10]. A modular computation for
inf P CO i (K) was obtained in [3].
Remark 2 If we introduce a new observation mask M̃i : Σ → ∆ ∪ {} such that M̃i = Mi Ti ,
we can use the results in [1, 10] to directly find the decentralized supervisors {S̃i } for G
(instead of using Theorem 1, which first finds the local supervisors {Si } for the local systems
{Gi }).
Our approach results in a computational saving in the setting of concurrent systems, and
also for general systems whenever the projections Ti ’s are observer maps [12]. Let us suppose
that the number of states in the global plant G is m, that in the ith local plant G i is mi , and
that in the automaton accepting the specification K is n. Then in the setting of concurrent
systems πi mi = m, and in the general setting mi ≤ m whenever Ti ’s are observer maps [13].
To obtain S̃i for G directly as in [1, 10], we need to determine inf P CO i (K), the infimal
prefix closed, (L(G), Σui )-controllable and (L(G), M̃i )-observable superlanguage of K. This
computation requires the global information L(G), and its generator can be shown to have
O(2n m) states. On the other hand, the approach of Theorem 1 computes the local supervisor
{Si } for local system {Gi } as the generator of inf P CO i (Ti (K)). This computation only
requires the local information L(Gi ), and can be shown to have O(2n mi ) states. Thus the
overall complexity of the direct approach of [1, 10] versus our approach is O(2 n |I|m) versus
O(2n
P
i
.
mi ), i.e., there is an improvement of a factor of P|I|m
mi
i
Having solved the “target specification” problem, we next consider the following “range
specification” problem:
Given two languages A and E with A ⊆ E ⊆ L(G), find local supervisors {Si , i ∈
I} such that A ⊆ L((
V
i
S̃i )/G) ⊆ E.
This problem should be viewed as a generalization of the “target specification” problem
studied above, since when A = E = K, it reduces to the target specification problem. The
following theorem provides a necessary and sufficient condition for the “range specification”
problem.
Theorem 2 Let G be a plant, Σ be the global event set, Σi ⊆ Σ, i ∈ I, be the local event
sets. Let Ti be the natural projection from Σ to Σi , Σci ⊆ Σi be the local controllable event
set, Mi be the local observation mask. Given two languages A and E with A ⊆ E ⊆ L(G),
there exist local supervisors {Si , i ∈ I} such that A ⊆ L((
L(G)
\
V
i
S̃i )/G) ⊆ E if and only if
(∩i Ti−1 [inf P CO i (Ti (A))]) ⊆ E,
where inf P CO i Ti (A) is the infimal prefix-closed, (L(Gi ), Σui )-controllable, and (L(Gi ), Mi )observable superlanguage of Ti (A).
Proof: The sufficiency is obvious, since we can choose the local supervisor Si such that
L(Si /Gi ) = inf P CO i (Ti (A)) for each ith local system Gi . For the necessity, if the local supervisors exist, then we let K = L((
V
i
T
S̃i )/G), i.e., K = L(G) (∩i Ti−1 [L(Si /Gi )]).
Then we have Ti (A) ⊆ Ti (K) ⊆ L(Si /Gi ). Since L(Si /Gi ) is prefix-closed, (L(Gi ), Σui )controllable, and (L(Gi ), Mi )-observable, and is also a superlanguage of Ti (A), we have
inf P CO i (Ti (A)) ⊆ L(Si /Gi ). Now since K ⊆ E, we obtain
L(G)
\
(∩i Ti−1 [inf P CO i (Ti (A))])
⊆ L(G)
\
(∩i Ti−1 [L(Si /Gi )])
=K⊆E
This completes the proof.
Remark 3 As in Remark 1, if the projections {Ti , i ∈ I} in Theorem 2 are identity projections, i.e., for each i ∈ I, Σi = Σ, then the necessary and sufficient condition in Theorem 2
becomes
∩i [inf P CO i (A)] ⊆ E.
From [3], we know this is equivalent to being inf P CCo(A) ⊆ E, where inf P CCo(A) is the
infimal prefix-closed, (L(G), Σu )-controllable, and (L(G), {Σui }, {Mi }) co-observable superlanguage of A, which is the result obtained in [10, Theorem 4.2]. So Theorem 2 extends the
result of [10].
Note, however, that by defining a new mask M̃i := Mi Ti , the result of [10] can be applied
to solve the problem addressed in Theorem 2. We present another approach in Theorem 2.
The advantages are same as those mentioned in Remark 2.
If A and E are given by the local specifications, then we can derive the following corollary
from Theorem 2:
T
T
Corollary 1 Let A = L(G) (∩i Ti−1 Ai ) and E = L(G) (∩i Ti−1 Ei ) with Ai ⊆ Ei ⊆ L(Gi ).
If inf P CO i (Ai ) ⊆ Ei for each i, then there exist local supervisors {Si , i ∈ I} such that
A ⊆ L((
^
S̃i )/G) ⊆ E.
i
Remark 4 In [7, Theorem 5], the authors studied the same problem for which Corollary 1
provides a solution. A sufficient condition was obtained under the assumption that M i ’s are
the projection type masks, and Σi = Σci ∪ Σoi . The sufficient condition implies the condition
we obtained in Corollary 1; plus we do not impose the assumption they impose. So we are
able to extend the result of [7].
4
Decentralized Local Control
Next we consider a special case of the problem in Theorem 1 as studied in [6]. This is
called the decentralized local control. In this setting the observation mask Mi for each local
supervisor is identity so that the local supervisor observes all the local events Σ i , and it
controls all the controllable events that are local, i.e., Σci = Σc ∩ Σi .
From Theorem 1 we have the following corollary:
Corollary 2 Under the same assumptions as in Theorem 1, and the additional assumption
that local observation masks Mi ’s are identity, given a nonempty prefix-closed language
K ⊆ L(G), there exist local supervisors {Si , i ∈ I} such that L((
if
K = L(G)
\
V
i
S̃i )/G) = K if and only
(∩i Ti−1 [inf P C i (Ti (K))]).
We show in the next lemma that the condition in Corollary 2 is equivalent to the condition
equivalent to the condition that K is globally controllable and decomposable when Σ ci =
Σc ∩ Σi , i ∈ I, where Σc = ∪i Σci is the global controllable event set.
Lemma 2
K = L(G)
\
(∩i Ti−1 [inf P C i (Ti (K))])
(1)
if and only if K is (L(G), Σu )-controllable and (G, {Ti , i ∈ I}) decomposable whenever for
each i ∈ I, Σci = Σc ∩ Σi , where Σu := Σ − Σc and Σc = ∪i Σci .
T
Proof: If K = L(G) (∩i Ti−1 [inf P C i (Ti (K))]), then from the definition of decomposability
we know K is (G, {Ti , i ∈ I}) decomposable. it is also easy to show that K is (L(G), Σu )controllable, so the necessity of the theorem holds.
For the sufficiency, if K is decomposable, then from Lemma 1, we have
K = L(G)
so
K ⊆ L(G)
\
\
(∩i Ti−1 Ti (K)),
(∩i Ti−1 [inf P C i (Ti (K))]).
To see the reverse containment, it suffices to show that non-zero length strings of
L(G)
\
(∩i Ti−1 [inf P C i (Ti (K))])
are in K, as the zero length string does belong to K(recall that K 6= ∅ and K = pr(K)).
T
Since it is easy to verify that L(G) (∩i Ti−1 [inf P C i (Ti (K))]) is prefix-closed, we only need
T
to show for s ∈ K, and σ ∈ Σ, if sσ ∈ L(G) (∩i Ti−1 [inf P C i (Ti (K))]) then sσ ∈ K. From
[4], we know inf P C i (Ti (K)) = Ti (K)(Σi − Σci )∗ ∩ Ti (L(G)), so
L(G)
\
(∩i Ti−1 [inf P C i (Ti (K))]) = L(G)
\
(∩i Ti−1 Ti (K)Ti−1 ((Σi − Σci )∗ )).
Now we have sσ ∈ Ti−1 Ti (K)Ti−1 ((Σi − Σci )∗ ), i ∈ I. If sσ ∈ Ti−1 Ti (K) for all i ∈ I, then
from the decomposability of K, sσ ∈ K. If for some i ∈ I, sσ 6∈ Ti−1 Ti (K), then we must
have σ ∈ Ti−1 ((Σi − Σci )∗ ). This implies either σ ∈ (Σi − Σci ) or σ 6∈ Σi . If σ 6∈ Σi , then
Ti (sσ) = Ti (s) ∈ Ti (K), implying sσ ∈ Ti−1 Ti (K), a contradiction. So we can only have
σ 6∈ Σci . Since Σci = Σc ∩ Σi and since σ ∈ Σi , σ 6∈ Σci implies σ 6∈ Σc , i.e., σ ∈ Σu . Since
sσ ∈ L(G), from the controllability of K, we have sσ ∈ K. This completes the proof.
Remark 5 The result of Lemma 2 supplements an earlier related result of Rudie-Wonham
[10, Proposition 4.3]. It was shown in [10, Proposition 4.3] that under the same conditions
as in Lemma 2, the condition of decomposability is equivalent to that of co-observability.
In contrast, we show that decomposability together with controllability of a language K is
equivalent to the condition of (1). In particular, condition (1) requires the knowledge of
only the local information since it requires the computation of inf P C i (Ti (K)) which can be
computed locally.
From Corollary 2 and Lemma 2, we have the following theorem:
Theorem 3 Let G be a plant, Σ be the global event set, Σi ⊆ Σ, i ∈ I, be the local event
sets. Let Ti be the natural projection from Σ to Σi , Σci ⊆ Σi be the local controllable event
set, Σc = ∪i Σci be the global controllable event set, Mi be the local observation mask. If
each observation mask Mi is identity and for each i ∈ I, Σci = Σc ∩ Σi , then for a given
nonempty prefix-closed sublanguage K of L(G), there exist local supervisors {Si , i ∈ I}
such that L((
V
i
S̃i )/G) = K if and only if K is (L(G), Σu )-controllable and (G, {Ti , i ∈ I})
decomposable, where S̃i is the extension of Si to the global system G, and Σu = Σ − Σc is
the global uncontrollable event set.
Remark 6 In [6], the following problem was considered:
Given a prefix-closed language E = ∩i Ti−1 (Ei ), let K = supP C(L(G)
T
E), find
conditions such that the supervisors {Si , i ∈ I} with L(Si /Gi ) = supP C i (Ei )
are globally optimal, i.e., L((
V
i
S̃i )/G) = K.
The authors gave a sufficient condition for the existence of the above special group of local
supervisors, i.e., the ones that are locally optimal. Theorem 3 gives a necessary and sufficient
condition for the existence of any kind of local supervisors.
5
Decentralized Control of Concurrent System
So far we have not assumed any structural properties of the system considered. Now we
suppose that the system G is composed of concurrent subsystems, Gi , with event sets Σi ,
i.e., G :=ki Gi , and Σ = ∪i Σi . From [11, Proposition 3.1], we have L(G) = ∩i Ti−1 [L(Gi )].
(Note that in the earlier analysis of sections 3 and 4 we also had the local systems {G i }
with L(Gi ) = Ti L(G), but we did not have the property that L(G) = ∩i Ti−1 [L(Gi )], which
holds for the concurrent system.) We also note that for the concurrent system the property
L(Gi ) ⊇ Ti L(G) holds instead of the definition L(Gi ) := Ti L(G) used for non-concurrent
systems. It can be seen that a concurrent system is a special case of the systems we considered
up to now. So we can derive the results for the concurrent system as before, by applying
L(G) = ∩i Ti−1 [L(Gi )]. We first give a result similar to that of Theorem 1, omitting the
proof.
Theorem 4 Let G be a concurrent system with G =ki Gi , i ∈ I, Σ be the global event set,
which is the union of local event sets, i.e., Σ = ∪i Σi , i ∈ I. Let Ti be the natural projection
from Σ to Σi , Σci ⊆ Σi be the local controllable event set, Mi be the local observation
mask. Then for a given nonempty prefix-closed sublanguage K of L(G), there exist local
supervisors {Si , i ∈ I} for {Gi , i ∈ I} such that ∩i Ti−1 [L(Si /Gi )] = K if and only if
K = ∩i Ti−1 [inf P CO i (Ti (K))],
where inf P CO i Ti (K) is the infimal prefix closed, (L(Gi ), Σui )-controllable, and (L(Gi ), Mi )observable superlanguage of Ti (K), and Σui := Σi − Σci is the local uncontrollable event set.
Remark 7 As we stated in Remark 2, we only need the local information L(Gi ) when
we design the local supervisor Si , since Si achieves the language inf P CO i (Ti (K)). On
the other hand to verify the condition in Theorem 1, we still need the global information
L(G). This is not the case for the concurrent systems as is obvious from the condition
of Theorem 4. So for the concurrent systems, we need not compute L(G). This provides
tremendous computational savings for large concurrent systems.
We also have a similar result as Theorem 3 for concurrent systems when each observation
mask Mi is identity, the proof of which is again omitted.
Theorem 5 Let G be a concurrent system with G =ki Gi , i ∈ I, Σ be the global event
set, which is the union of local event sets, i.e., Σ = ∪i Σi , i ∈ I. Let Ti be the natural
projection from Σ to Σi , Σci ⊆ Σi be the local controllable event set, Σc = ∪i Σci be the
global controllable event set, Σu = Σ − Σc be the global uncontrollable event set, Mi be
the local observation mask. If Mi is identity for each i and Σci = Σc ∩ Σi , then for a given
nonempty prefix-closed sublanguage K of L(G), there exist local supervisors {Si , i ∈ I} such
that ∩i Ti−1 [L(Si /Gi )] = K if and only if K is (L(G), Σu )-controllable and (G, {Ti , i ∈ I})
decomposable.
Remark 8 Note that since now we have L(G) = ∩i Ti−1 [L(Gi )], the decomposability is the
same as the separability defined in [11, Definition 2.1]. In [11, Theorem 4.1], the authors
studied a similar problem as studied in Theorem 5:
Given a prefix-closed language E ⊆ Σ∗ , let K = supP C(L(G) ∩ E), find the local
supervisors {Si , i ∈ I} such that ∩i Ti−1 [L(Si /Gi )] = K.
Under the assumption that (∀i 6= j)Σui ∩ Σj = ∅, the authors gave the same necessary and
sufficient condition as Theorem 5 for the existence of the local supervisors. In Theorem 5,
we need that Σci = Σc ∩ Σi . This is equivalent to (∀i 6= j)Σui ∩ Σcj = ∅, which is weaker
than the assumption in [11]. So our result extends that of [11].
For concurrent systems, results similar to those of Theorem 2 and Corollary 1 can also
be obtained, which we state below:
Theorem 6 Let G be a concurrent system with G =ki Gi , i ∈ I, Σ be the global event set,
which is the union of local event sets, i.e., Σ = ∪i Σi , i ∈ I. Let Ti be the natural projection
from Σ to Σi , Σci be the local controllable event set, Mi be the local observation mask. Given
two languages A and E with A ⊆ E ⊆ L(G), there exist local supervisors {S i , i ∈ I} such
that A ⊆ ∩i Ti−1 [L(Si /Gi )] ⊆ E if and only if
∩i Ti−1 [inf P CO i (Ti (A))] ⊆ E.
Corollary 3 Let G be a concurrent system with G =ki Gi , i ∈ I, A = ∩i Ti−1 Ai and
E = ∩i Ti−1 Ei with Ai ⊆ Ei ⊆ L(Gi ). If inf P CO i (Ai ) ⊆ Ei for each i ∈ I, then there exist
local supervisors {Si , i ∈ I} such that
A ⊆ ∩i Ti−1 [L(Si /Gi )] ⊆ E.
6
Illustrative example
In this section, we apply our results to a simple manufacturing system which is a modified
version of that studied in [7].
α1
Mn1
β1
H
τ1
τ2
B
α2
Mn2
β2
Figure 2: A manufacturing system
The system, shown in Figure 2, consists of two machines {Mn1 , Mn2 }, one material
handling device H of unit capacity, and one buffer B of capacity 2. We assume that there
is an infinite number of workpieces at the input port of Mn1 , and a workpiece is taken away
from Mn2 to the output port after the completion of its processing in the system.
The event labels in Figure 2 represent the following actions:
α1 : a workpiece is taken by Mn1 from the input port;
α2 : a workpiece is taken by Mn2 from B;
β1 : a workpiece is taken by H from Mn1 after the completion of its processing;
β2 : a workpiece is taken away from Mn2 to the output port;
τ1 : a workpiece is sent to B by H;
τ2 : a workpiece is sent to Mn1 from Mn2 for re-processing.
The event τ1 is the only uncontrollable event. At first, we consider the case that all events
are observable.
α1
τ2
α2
α2
β2
β1
(a)
τ2
β1
α2
τ1
(b)
(c)
Figure 3: Subsystems G1 , G2 , and G3
The uncontrolled system is given as
G = G1 k G2 k G3 ,
where G1 and G2 represent Mn1 and Mn2 respectively, and are shown in Figure 3(a) and
2(b), G3 is the subsystem of the combination of H1 and B1 , and is shown in Figure 3(c). Here
we consider only closed languages, therefore all states are considered to be marked states.
The global event set is Σ = {α1 , α2 , β1 , β2 , τ1 , τ2 }, and the local event sets are Σ1 =
{α1 , β1 , τ2 }, Σ2 = {α2 , β2 , τ2 }, and Σ3 = {β1 , τ1 , α2 }. We have Σ1c = Σ1 , Σ2c = Σ2 , Σ3c =
{β1 , α2 }, Ti : Σ → Σi , Mi = Id, for i = 1, 2, 3.
α2
β
τ1
1
α2
α2
β1
τ1
β
1
α2
Figure 4: Automaton for E3
The range specifications are given as A = ∩i Ti−1 (Ai ) and E = ∩i Ti−1 (Ei ), where E1 =
A1 = (α1 β1 τ2 β1 )∗ , E2 = A2 = (α2 β2 τ2 β2 )∗ , these two local specifications together require
that a workpiece being processed twice before leaving the system and both machines process
the new and old workpieces in turn. A3 = (β1 τ1 α2 )∗ , which requires that the local system
G3 can run continuously. E3 is generated by the automaton in Figure 4, requiring that the
overflow and the underflow of the combined H and B system be avoided.
It is easy to check that
inf P CO 3 (A3 ) = A3 ⊆ E3 ,
along with the fact that
inf P CO i (Ai ) = Ai = Ei , i = 1, 2.
So from Corollary 3 we know that there exist local supervisors {Si , i = 1, 2, 3} such that
A ⊆ ∩i Ti−1 [L(Si /Gi )] ⊆ E.
In fact, each Si can be chosen as the deterministic automaton that achieves the language
inf P CO i (Ai ) = Ai .
Now suppose that, owing to a failure in a sensor, event α2 becomes unobservable in G3 ,
i.e., M3 (α2 ) = . Then we can calculate inf P CO 3 (A3 ), which is the language generated by
the automaton in Figure 5.
β
τ1
1
α2
β
1
τ1
Figure 5: Automaton for inf P CO 3 (A3 )
From Figure 4 and Figure 5, it can be seen that inf P CO 3 (A3 ) 6⊆ E3 , since (β1 τ1 )3 ∈
inf P CO 3 (A3 ) and (β1 τ1 )3 6∈ E3 . So the condition in Corollary 3 is not satisfied. Since this
condition is not necessary for the existence of the local supervisors, we would still like to
know whether we can somehow verify the existence of local supervisors for this case. For
this we check the condition of Theorem 6.
We first obtain the global range specification A = ∩i Ti−1 (Ai ) and E = ∩i Ti−1 (Ei ), where
the automaton generating the language A is shown in Figure 6 (the automaton for E is
omitted). Next we compute inf P CO i [Ti (A)]. We have the following results:
inf P CO 1 [T1 (A)] = A1
inf P CO 2 [T2 (A)] = A2
inf P CO 3 [T3 (A)] = inf P CO 3 (A3 ).
α1
β
2
α2
β
τ1
β1 β2
τ1
1
β
α1
2
α1
α2
τ1
β
α2
2
α1
τ2
β
1
τ1
Figure 6: Automaton for A
Further, we can also verify that ∩i Ti−1 [inf P CO i (Ti (A))] ⊆ E (the details are omitted here).
So the condition in Theorem 6 does hold, which implies that local supervisors exist for
this case. Here {S1 , S2 , S3 } can be chosen as the deterministic automatons which achieve
languages {A1 , A2 , inf P CO 3 (A3 )} respectively. It can be seen that S3 alone cannot prevent
the overflow in G3 , since it enables (β1 τ1 )k for any finite k. But S2 and S3 together prevent
the overflow in G3 , which is guaranteed from ∩i Ti−1 [inf P CO i (Ti (A))] ⊆ E.
If we introduce a new mask M30 = M3 T3 , then we can also use the results in [1, 10] to solve
the above problem. But for this approach we need to first compute L(G), and next compute
inf P CO i [A] with respect to {L(G), Σui , Mi0 }, where M10 = T1 , M20 = T2 , and Σui = Σ − Σci .
Next we need to check whether the condition ∩i [inf P CO i (A)] ⊆ E is satisfied. For the above
problem, we can show that this condition is satisfied, and we can choose the decentralized
supervisors {S̃i , i = 1, 2, 3} to be the deterministic automaton which achieve the languages
{inf P CO i (A), i = 1, 2, 3} respectively. As mentioned in Remark 2, our approach yields an
improvement in the computational complexity by a factor of P|I|m
when compared to the
mi
i
approach in [1, 10] (m is the number of states in G, mi is the number of states in Gi , and
|I| is the total number of systems). In the present example, m = m1 × m2 × m3 = 23 = 8,
and |I| = 3. So the improvement factor is 4. There is another difference, namely, the
decentralized supervisors {S̃i , i = 1, 2, 3} derived from the approach in [1, 10] control the
global system G directly, however, the local supervisors {Si , i = 1, 2, 3} derived from our
approach only control each local system {Gi , i = 1, 2, 3} respectively.
Finally let us consider a small variation and change the buffer capacity from 2 to 1,
and continue to assume that event α2 is unobservable in G3 . We change E3 , requiring that
the overflow and the underflow of the combined H and B system be avoided, accordingly.
The modified E3 is the language generated by the automaton in Figure 7; the other specifications remain unaltered. Then by using the same method as above, we can verify that
∩i Ti−1 [inf P CO i (Ti (A))] ⊃ E (the details are omitted). So the condition in Theorem 6
α2
β1
τ1
β1
α2
Figure 7: Automaton for new E3
does not hold, which implies that local supervisors do not exist when the buffer capacity is
reduced to one.
7
Conclusion
In this paper we studied the decentralized supervisory control problem of concurrent discrete
event systems under partial observation. The main result is the derivation of a necessary
and sufficient condition for the existence of decentralized supervisors that ensure that the
controlled behavior of the system lies in a given range. This result was specialized to the
settings of decentralized local control and decentralized control of concurrent systems. These
extended the results of [1, 10, 6, 7, 11] on the decentralized control of concurrent systems under partial observation. The ideas were illustrated by application to a simple manufacturing
system.
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